OF SIMULTANEOUS ORDINARY DIFFERENTIAL EQUATIONS. 525 
and (V.) respectively are substituted in (I.), the results are respectively equivalent to 
(II.) and (IV.). Hence (II.) and (IV.) are also made equivalent, and therefore (IV.) 
may be derived from (II.) by substituting appropriate functions of « for c,,... C. 
Hence, by subtracting (III.) from the derivative of (II.), we have 
» OF, de 
Se eae see / 
2s = 0 (EY 298 0) anf ere (AVAL) 
Thus, unless ¢,, c,... ¢, are all constants, 
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Bi SN Regeigt fae al ce tesla came nas GN GLELG)g 
§ 3. The equations (II.) and (VII.) may be considered as defining @, y,,...y, in 
terms of ¢),¢)... ¢,, and the equations (VI.) will then give the ratios de, :de,:...:de, 
in terms of ¢, ¢,...¢, We thus have x — 1 differential equations connecting the 
nm quantities c,,...¢,. Their complete primitive will involve 1 — 1 arbitrary constants, 
and by eliminating c,...c, from (II.) and (VII.) and this complete primitive we 
have a solution of (1.) involving n — 1 arbitrary constants, which we may call the 
Jirst singular solution of (1.) 
The differential equations given by (VI.) may havea first singular solution involving 
7. — 2 arbitrary constants, and from this we should derive the second singular solution 
of (I.) by eliminating c,... ¢, as before. 
This process may go on till we have n singular solutions, with » — 1,n — 2... 2, 1,0 
arbitrary constants respectively, or it may stop at any stage. If, for instance, the 
left-hand side of the equation (VII.) were an absolute constant, there would be no 
singular solution at all. 
Formation of Singular Solutions from the Differential Equations. 
§ 4. The equation (VII.) is the condition that two solutions of (II.) when solved for 
€), Cy... Shall coincide. But, generally, when this happens, the equations (III.) 
give coincident sets of values for p;, DP... + Pu 
of p, ... pn from (Y.), and become relations (A) connecting ¢), cz, ... ¢, with x, and yj, y,... yn which 
are given as functions of = by (IV.). 
Now by substituting the values of p,...p, from (III.) in (1.), we have integral equations which must 
be included in (I1.). 
The number of these equations will be » — m, because m ave already satisfied identically. 
These equations are also satisfied if (A) and (IV.) are; for the values of p,...p, given by (III.) and 
(Y.) are then the same and those given by (V.) satisfy (1.). 
The system (II.) will supply exactly m further equations which in combination with (A) and (IV.) 
give the values of c, c),...c, in terms of a. 
